This invention relates to a method and apparatus for measuring changes in shapes and wavelength (frequency) at various sections of ultrashort optical pulses. In the past, these changes have not been capable of being measured because they change too rapidly in a very short time. Because this time is close to or less than the response time of currently available optical detectors, these detectors could not measure these changes. More particularly, the invention relates to a method and apparatus for measuring ultrashort optical pulses, which measures in detail both an intensity shape and a phase shape of pulses which correspond to an integration of a change of instantaneous frequency (chirping) when the optical pulses form pulse sequences that periodically repeat at a fixed time interval.
FIG. 1 is a schematic diagram of a conventional, prior art, measuring device for measuring ultrashort optical pulses. A pulsed light beam to be measured 100 is shown incident on beam splitter 24. A portion of the pulsed light beam to be measured 100 is reflected by beam splitter 24 to enter a spectrum measurement system 102, and the rest of the beam enters an intensity autocorrelation measurement system 104. The spectrum measurement system 102 has a high-resolution spectroscope 26 mounted with a wavelength scanner (a scanning mechanism) 27, a converging lens 25 which focuses the light beam at the incident slit 106 of the spectroscope 26, an optical detector 28, and an integrating amplifier 29 which amplifies the output voltage from the optical detector 228.
In the intensity autocorrelation measurement system 104, the incident light beam is divided into two light beams by a further beam splitter 1. One of the split beams is reflected by a fixed prism 2 while the other beam is reflected by a movable prism 3, which is movable along the optical axis by a mover 4. These two reflected beams are recombined by the beam splitter 1. The optical system comprising the beam splitter 1 and two prisms 2, 3 is called a Michelson interferometer.
Assuming an electric field of the light before entering into the Michelson interferometer is denoted as E(t) as a function of the time t, the electric field after the recombination can be expressed as: EQU [E(t)+E(t-.tau.)]/2
where the delay time .tau. is a time delay due to the path difference. This amount can be obtained by dividing a relative path length difference l, between the path extending through the prism 2 and the path extending through the prism 3, by a light velocity c. Since the prism 3 can be moved by mover 4, .tau. can be changed arbitrarily.
After recombination, the light beam is focused by lens 5 at a doubling crystal 7. This crystal 7 generates second harmonic light, that is. light having a wavelength of one half of that of the incident light. The electric field, E.sub.2 (t), of this second harmonic light is proportional to [E(t)+E(t-.tau.)].sup.2. The output light from the crystal 7 is focused by a lens 6 at an optical detector 9. However, optical filter 8 is disposed between lens 6 and detector 9, and allows only the second harmonic light to pass therethrough and to reach the optical detector 9. Since the response time of the optical detector 9 is significantly longer than the duration of the optical pulses to be measured, and of the second harmonic pulses generated from the crystal 7, it cannot accurately detect the shapes of the second harmonic light pulses. Therefore, the pulse shape of an output voltage from the detector 9 has inherent inaccuracies as compared with what the voltage should be. A time integration of the pulse shapes, i.e. their area, however, is proportional to the energy of the second harmonic light pulses. The output voltage pulses from the detector 9 are converted into a constant voltage proportional to the area of said output voltage pulses by an integrating amplifier 10 with a large time constant T.
The operation of the conventional device to measure ultrashort pulses is described below.
First, a signal selector switch 31 is switched to a first position to connect an A/D converter 32 to the output of spectrum measuring system 102. The central wavelength of spectroscope 26 is then scanned at a constant rate by wavelength scanner 27, while the output voltage of the detector 28 is read-in and stored time-sequentially in a computer 18 via A/D converter 32. In this way, a data sequence representing spectrum D as a function of wavelength, or D(.lambda.), of the optical pulses is obtained by this first measurement. For a second measurement, signal selector switch 31 is switched to a second position to connect A/D converter 32 to intensity autocorrelation measurement system 104. While mover 4 moves prism 3 at a constant speed, the output voltage from integrating amplifier 10 is read-in and stored time-sequentially in computer 18 via the A/D converter 32. A cutoff frequency, which is the reciprocal of the time constant T of the integrating amplifier 10, is set sufficiently lower than a central frequency f.sub.0 of the optical pulses. This central frequency f.sub.0 is determined by the central wavelength .lambda..sub.0 of the optical pulses to be measured, and the speed of mover 4, using the equation: EQU f.sub.0 =2V/.lambda..sub.0 ( 1)
The data sequence thus obtained is proportional to 1+2G.sub.2 (.tau.), where G.sub.2 (.tau.) is an intensity autocorrelation and can be expressed as a function of the intensity shape of pulses I(t) as follows: ##EQU1##
The measurement process ends at this point, and the analyzing process begins.
The spectrum D(.lambda.) is then converted to a spectrum D(.omega.) as a function of frequency w. When an electric field of the object pulses is denoted as E(t), and its Fourier transformation as E(.omega.), the absolute value of E(.omega.), i.e. .vertline.E(.omega.).vertline., can be obtained from a square root of D(.omega.) as follows: ##EQU2## The electric field of the object pulses E(t) can be completely reconstructed from a phase .phi.(.omega.) in the Fourier component of the relation: EQU E(.omega.)=.vertline.E(.omega.) exp [i.phi.(.omega.)] (4)
(where i is an imaginary unit) Then, .phi.(.omega.) is expanded around the central wavelength .omega..sub.0 of the spectrum as: EQU .phi.(.omega.)=.phi..sub.0 +(.omega.-.omega..sub.0)p+(.omega.-.omega..sub.0).sup.2 q+(.omega.-.omega..sub.0).sup.3 r (5)
where .phi..sub.0 denotes an arbitrary constant, and p only provides a correction for the central frequency .omega..sub.0 of the spectrum, and does not affect the intensity shape. Therefore, only the expansion coefficients q and r can be determined by this method. In order to obtain the values of q and r, the known, least-squares method is employed and values are adjusted until G.sub.tr (.tau.), calculated according to the two equations below, coincides with an actually measured intensity autocorrelation as closely as possible. ##EQU3## By this method, .phi.(.omega.) is approximated with a precision as high as third order. After calculating E.sub.tr (t) from the above equation, the intensity shape and the phase shape of the measured pulses can be calculated from .vertline.E.sub.tr (t).sup.2 .vertline. and arg[E.sub.tr (t)] respectively. The instantaneous frequency and wavelength at various points on pulses can be obtained from time derivatives of their phase shape.
However, although this method can provide an analysis of coefficients of .phi.(.omega.) with accuracy to third-order, this method cannot reproduce in detail the intensity shape and phase shape of arbitrarily shaped pulses. Although the situation can be improved by obtaining a higher-order expansion coefficient of .phi.(.omega.), this is difficult and impractical, as the calculation time would be greatly increased.
Another great difficulty with this prior art method arises from the requirement for proper spectrum measurement of a spectroscope 26 of extremely high resolution and precision. For instance, the full width at half maximum (FWHM) of spectrum is around 5 .ANG. with a lps wide optical-pulse of 0.6 .mu.m central wavelength.
Using this method to record fifty measuring points which are reasonably independent from each other within the FWHM, the spectroscope would require a resolution of less than 0.1 .ANG.. If the width of a measured pulse becomes longer, the resolution must be improved proportionally.
Devices which meet these specifications include a diffraction grating spectrometer having a focal length longer than 1 m, and a scanning Fabry-Perot etalon. Neither of them, however, could be used to provide a simple measuring method. The former spectroscope is large in size and has a difficult-to-adjust optical alignment. The latter spectroscope requires even more careful optical alignment than the former.
As stated above, in the conventional methods of measuring ultrashort optical pulses, both a high-resolution spectroscope (for spectrum measurement) and a separate Michelson interferometer (to measure intensity autocorrelation) are needed. These are both expensive and cumbersome items. The spectroscope is especially expensive, in addition to being large and bulky, and is difficult to align. Therefore, the spectroscope has accounted for a high percentage of the cost and volume of a measuring system in the prior art, presenting a formidable problem in miniaturizing the system, reducing the cost, and simplifying adjustments such as alignment. Moreover, as the data analysis method using this structure is still a rough approximation, optical pulses could not be evaluated heretofore in detail.
The object of this invention is to obviate the aforementioned problems encountered in the prior art and to provide an ultrashort optical pulse measurement method which can measure optical pulses in detail without the necessity for a high-resolution spectroscope which makes the system bulky and costly.
The first aspect according to this invention measures ultrashort pulses repeating with a constant period. More particularly, the method for ultrashort pulse measurement described splits a measured light beam into two beams using a beam splitter. The two beams follow different paths which have a relative path difference therebetween by directing the two beams to follow different paths. The beams are recombined, and the combined beam is focused on a doubling crystal. The intensity of the second-harmonic light generated in the doubling crystal is converted to a voltage by an optical detector.
According to the first method, the changes in intensity of the second-harmonic light due to the relative path length difference are recorded, and the data thus obtained is Fourier-analyzed to obtain three spectra which are distributed at a frequency interval of the interference fringes of the fundamental wave. The intensity shape of the measured pulse is reconstructed using an inverse Fourier transformation of a spectrum closest to zero frequency among those spectra, by taking the square root of this spectrum. The phase of the measured pulse is reconstructed by iterative calculations using the square root of the spectrum closest to a frequency twice as high as that of the interference fringe of the fundamental wave and using the reconstructed intensity shape.
The second method of this invention measures ultrashort pulses repeating with a constant period. More particularly, a method for ultrashort pulse measurement wherein a measured light beam is split into two beams by a beam splitter is described. A relative path difference is provided between the two beams by letting the beams pass through different paths. Subsequently the beams are recombined, and the combined beam is focused on and enters a doubling crystal. The intensity of the second-harmonic light generated in the crystal is converted to a voltage by an optical detector. The output voltage from the optical detector is divided into two portions according to this method. One of the portions passes through a low-pass filter in order to extract a frequency component near DC. The other portion passes through a high-pass filter in order to extract a frequency component of a frequency double the measured frequency in correspondence with the frequency of the interference fringe of the fundamental wave. An amplitude of the component from the high-pass filter is obtained using an AC voltmeter. The DC and AC voltages thus obtained are recorded in time sequence while varying said relative path length difference at a constant rate, to produce a plurality of paired data sets. The data on the component near CC is Fourier-analysed, a square root is taken of the obtained spectrum, and the intensity shape of the measured pulse is reconstructed by its inverse Fourier transformation. The data on the AC voltage value are Fourier-analysed, its square root is taken, and from this resultant spectrum and the intensity shape, the phase of the measured pulse is reconstructed by iterative calculations.
Using this novel technique, autocorrelation of a second-harmonic electric field, instead of a spectrum, is performed. More particularly, these methods simultaneously measure and record the intensity autocorrelation and the autocorrelation of the second-harmonic electric field of a pulse using a Michelson interferometer and a doubling crystal as the operative components. Without the necessity for any spectroscope, the intensity shape as well as the phase shape of an optical pulse can be measured.
According to these methods, the light beams having relative path differences from traversal of a Michelson interferometer are recombined, and the resultant beam is focused, by a lens, on the doubling crystal. The second-harmonic light thus generated is transformed into a voltage by an optical detector.
The voltage from the optical detector under this condition is proportional to the formula (8): EQU 1+2G.sub.2 (.tau.)+4F.sub.1 (.tau.) cos [.omega..sub.0 .tau.+.phi..sub.1 (.tau.).vertline.+F.sub.2 (.tau.) cos [2.omega..sub.0 .tau.+.phi..sub.2 (.tau.)] (8)
The first and second methods according to this invention are characterized in that G.sub.2 (.tau.) and F.sub.2 (.tau.) of the above formula are measured respectively, where .tau. denotes a delay time and .omega..sub.0 =2.pi.c/.lambda..sub.0 is the frequency of the central angular frequency of the measured pulse. F.sub.2 (.tau.) and 0.sub.2 (.tau.) are defined by the equation below. ##EQU4## when the electric field of the pulse E(t) is expressed by the equation: EQU E(t)=[.epsilon.(5) exp (-.omega..sub.0 t)+.epsilon.*(t) exp (i.omega..sub.0 t)]/2 (9)
where the asterisk (*) denotes a complex conjugate. The left-hand side of the equation (10) is called an autocorrelation of a second-harmonic electric field.
While the prior art methods measure only the G.sub.2 (.tau.) term in equation (8) using a Michelson interferometer, the present invention measures both G.sub.2 (.tau.) and F.sub.2 (.tau.) using equation (8), thereby eliminating the necessity for an additional spectrum measurement with a spectroscope.
In the analyzing stage of calculating intensity shapes I(t) and .phi.(t) from the two autocorrelated shapes mentioned above, both the first and the second invention method perform analyses on the assumption that the pulses are symmetric pulses, which are symmetrical about a certain time point. Under this assumption, the intensity shape of the pulse I(t) is reconstructed from the intensity autocorrelation shape data G.sub.2 (.tau.) in accordance with equation (11) below: ##EQU5## This calculation shows that, when pulses are symmetrical, no matter how complicated their shapes are, they can be reproduced. A Fourier transformation on an autocorrelation data F.sub.2 (.tau.) of the second-harmonic electric field can give the square of the absolute value of the Fourier transformed value of the square of the electric field of the measured pulse. In other words, using an assumption. of symmetrical pulses, the relation below holds: ##EQU6## On the other hand, [.epsilon.(t)].sup.2 can be expressed using the phase .phi.(t) of the pulse and the intensity shape I(t) as: EQU [.epsilon.(t)].sup.2 =I(t) exp [2i.phi.(t)]
If the above is combined with the definition of S(.omega.), the relation below holds: ##EQU7## If .vertline.S(.omega.).vertline. is calculated from data F.sub.2 (.tau.) using equation (12), and I(t) is already obtained, then the phase .phi.(t) of the pulse can be obtained with a high precision by iteratively computing equation (14): EQU exp (2i.phi..sub.k+1)=arg{F.vertline.S.vertline.arg (F[I exp (2i.phi..sub.k)])} (14)
where the letter F denotes a Fourier transform, and arg denotes a calculation of the phase of a complex number. The method of calculation of the equation (14) is described in detail by Gerchberg et al. in OPTIK vol. 35, p. 237.
Unlike the conventional approximate solution, which is limited to a finite order, the above analytical method, in principle, takes into consideration up to an infinite order, and can calculate the intensity shape and phase of a measured pulse at an arbitrarily high precision.
When a second-harmonic electric field is expressed as a function of time t as u(t), a square root of the spectrum close to a frequency double the interference fringe of the fundamental wave is an absolute value .vertline.u(.omega.).vertline. of a Fourier-transformed u(t), i.e u(.omega.). A square root of the spectrum close to zero frequency (DC) is an absolute value .vertline.I(.omega.).vertline. of the Fourier-transformed intensity shape I(t)=.vertline.u(t).vertline., i.e. I(.omega.). If u(t) is a symmetric pulse about a given time point, .vertline.u(.omega.).vertline. and .vertline.I(.omega.).vertline. described above provide sufficient information to obtain the intensity shape I(t) and phase shape (1/2)arg[u(t)]. If it is not a symmetric pulse, however, this is not sufficient information to accurately reconstruct the pulse.
In addition to the dependence of the measured second-harmonic intensity on the relative path difference, if the spectrum of the measured pulse is obtained, this data will be sufficient to reconstruct the asymmetric pulses. However, if a high resolution spectroscope is introduced for the measurement of the spectrum, the system will become bulky in size and expensive in cost, failing to achieve the purpose of this invention.
In order to solve this dilemma, the present inventors conceived a measurement method in which an optical detector was newly added to the above system to measure the intensity of a fundamental wave light without a spectroscope. If the output from this optical detector is recorded in time sequence while varying the relative path difference, it is essentially measuring what is otherwise obtained by performing an inverse Fourier transformation on the spectrum of the measured pulse (which is called an interferogram). When the obtained data is Fourier-transformed, then its spectrum is obtained.
The third method according to this invention comprises the steps of splitting an object light beam into two beams, varying the relative path difference of the split paths, recombining those two light beams, making the combined beam enter a crystal with a doubling ability to generate second-harmonic light, converting the intensities of the fundamental wave and the second-harmonic light into electric signals proportional thereto, recording the changes in intensity of the fundamental light and the second-harmonic light due to the relative path difference, and performing Fourier-analysis on the recorded data to calculate an intensity shape and a phase shape of the measured pulse.
In the calculation performed according to this third method, a Fourier-transformation of an electric field of the measured pulse is obtained as the spectrum close to the fundamental frequency .omega..sub.0, by Fourier-analyzing the intensity-change data of the fundamental light. A Fourier-transformation of a second-harmonic electric field is obtained as a spectrum close to 2.omega..sub.0 by Fourier-analyzing the intensity change data of second-harmonic light. A Fourier-transformation of the intensity of the measured optical pulse is obtained as a spectrum close to zero frequency DC. Based on these three Fourier-transformations, iterative calculations are performed.
This method measures the intensities of not only the second-harmonics but also of the fundamental wave light having the same wavelength as the original optical pulse using separate optical detectors to convert them into voltages.
The output voltage of the optical detector which measures the intensity of the fundamental light can be expressed as follows: EQU 1+Re[G.sub.1 (.tau.) exp (-i.omega..sub.0 .tau.)] (15)
wherein .tau. denotes a delay time, .omega..sub.0 =2.pi.c/.lambda..sub.0 the central angular frequency of the measured pulse, and the symbol Re denotes taking the real part of a complex number. If the electric field E.sub.0 (t) is represented as: EQU E.sub.0 (t)=Re[E(t) exp (-i.omega..sub.0 .tau.)] (16)
G.sub.1 (.tau.) is referred to as an electric autocorrelation function and expressed as below. ##EQU8## When the data G.sub.1 (.tau.) is Fourier-transformed, the square of the absolute value of the Fourier-transform E(.omega.) of the electric field E(t) of the measured pulse is obtained, i.e. ##EQU9## The right-hand side of the above equation expresses the spectrum of the optical pulse. When a spectrum of light is measured by an interferometer instead of a spectroscope, this method is called Fourier-transform spectroscopy.
The output voltage of the optical detector which measures the intensity of the second-harmonics is proportional to equation (19) EQU 1+2G.sub.2 (.tau.)+4Re[F.sub.1 (.tau.) exp (i.omega..sub.0 .tau.)]+Re[F.sub.2 (.tau.) exp (2i.omega..sub.0 .tau.)] (19)
The intensity I(t) of the optical pulse and the electric field of the second-harmonics u(t) are defined as follows: EQU I(t)=E(t)E*(t) (20) EQU u(t)=E(t)E(t) (21)
Equations (22) and (23) define how I(t), u(t), the intensity correlation function G.sub.2 (.tau.) and the correlation function of the second-harmonic electric field F.sub.2 (.tau.) can be used to obtain the desired values as below. ##EQU10## By Fourier-transforming the above, the square of the absolute value of the Fourier-transform I(.omega.) of the intensity I(t) of each optical pulse, and the square of the absolute value of Fourier-transform u(.omega.) of the second-harmonic electric field u(t) are obtained as follows. ##EQU11## This inventor of the present invention has conceived a Fourier analyses what is obtained by subtracting a flat DC component from the second-harmonic intensity expressed by equation (19). FIG. 11A shows the signals expressed by equation (19), with FIG. 11C showing the Fourier-transformed values thereof. The cycle of oscillation apparent from the signals coincides with the central wavelength (.lambda..sub.0) of the measured light. This corresponds to the period of the optical oscillation of the measured light when the relative path difference is divided by the light velocity, and could be substituted by a delay time.
If this signal is Fourier-analysed, three peaks appear as shown in FIG. 11C. The middle peak is due to an apparent oscillating component in the signals and located close to the frequency .omega..sub.0 of the optical oscillation of the measured light. This frequency .omega..sub.0 is the frequency of an interference fringe of the fundamental wave. A result of the Fourier-analysis shows other peaks near zero frequency and near 2.omega..sub.0. These three peaks originate from the three terms in equation (19). G.sub.2 (.tau.), F.sub.1 (.tau.) and F.sub.2 (.tau.) of equation (19) can be easily separated from each other by performing a Fourier-analysis on the whole signals of the equation (19) as they are differentiated by a carrier wave frequency .omega..sub.0 which varies more quickly than those three. .vertline.I(.omega.).vertline..sup.2 of equation (24) is obtained from a peak close to zero frequency, while .vertline.u(.omega.).vertline..sup.2 of equation (25) is obtained by extracting a peak near 2.omega..sub.0 and translating it by -2.omega..sub.0.
By performing Fourier-analysis on the intensity signals of the fundamental wave light from which a flat DC component has been subtracted, a peak centered at .omega..sub.0 is obtained as shown in FIG. 11D. By translating this peak by -.omega..sub.0, .vertline.E(107 ).vertline..sup.2 of equation (18) is obtained.
As shown above, when a Fourier-analysis is performed on the data indicative of the dependence of the intensities of the fundamental wave light and second-harmonic light on relative path difference (delay time), the quantities (functions) below are obtained: EQU .vertline.I(107 ).vertline..sup.2, .vertline.u(.omega.).vertline..sup.2, .vertline.E(.omega.).vertline..sup.2
Using thus obtained .vertline.I(107 ).vertline., .vertline.u(.omega.).vertline., and .vertline.E(.omega.).vertline., calculations are iteratively performed to reconstruct the original optical pulse. FIG. 12 shows this calculation method, wherein the letters F.T. denote a Fourier-transformation, I.F.T. denote an inverse Fourier-transformation and csqrt denotes calculation of a square root of a complex number. The reconstruction of a pulse can be reduced, in principle, to obtain the phase of a Fourier-transformation E(.omega.) of an electric field E(t). This is because as the absolute value .vertline.E(.omega.).vertline. of the Fourier-transformation has been already obtained from the measured data, the Fourier-transformation E(.omega.) will be known once its phase is known. The electric field E(t) can be obtained by its inverse F transformation. The intensity shape of the measured pulse is obtained from .vertline.E(t).vertline..sup.2 while the phase shape of the pulse is obtained from arg{E(t)}. The iterative calculation of this invention utilizes the fact that the relationship between equations (20) and (21) hold between E(t), I(t), and u(t), and can be used to obtain the phases of .vertline.E(.omega.).vertline., .vertline.I(.omega.).vertline., and .vertline.u(.omega.).vertline.. More specifically, an initial preparation is made by giving a random phase to .vertline.E(.omega.).vertline.. Using E(t), obtained by its Fourier-transformation, calculation is iteratively performed from the lower left of FIG. 12. I(t) and u(t) are calculated from this E(t) according to equations (20) and (21). I(t) and u(t) are then respectively Fourier-transformed. The absolute values .vertline.I(.omega.).vertline. and .vertline.u(.omega.).vertline. of thus obtained I(.omega.) and u(.omega.) are substituted for .vertline.I(.omega.).vertline. and .vertline.u(.omega.).vertline. which have been obtained by Fourier-analysis of the measured data at the previous stage. The difference between the values .vertline.I(.omega.).vertline. or .vertline.u(.omega.).vertline. before and after the substitution is used for judging the convergence. When there is no longer any difference between these values, the pulse can be considered to be fully reconstructed. By performing an inverse Fourier-transformation on I(.omega.) and u(.omega.), they are converted to I(t) and u(t), and the absolute value of u(t) is replaced by thus obtained I(t). A complex square root of u(t) is taken to convert to E(t). The E(t) is further Fourier-transformed to obtain E(.omega.), and its absolute value is substituted for the known .vertline.E(.omega.).vertline.. The difference of values of .vertline.E(.omega.).vertline. between before an after substitution is used for judging convergence similarly to the cases of I(.omega.) and u(.omega.). After substitution, E(t) will be obtained by an inverse Fourier-transformation on (.omega.), to return to the starting point.
The E(t) which is obtained by completing this calculation loop provides a value of E(t) closer to the electric field of the measured optical pulse than the original E(t). The degree of its approximation is reflected in the differences of absolute values of three types of Fourier-transforms between before and after the substitutions. The calculation therefore should be repeated until the differences become sufficiently small.
The analysing method described as the third method, unlike the first and the second methods of this invention, is not limited in effectiveness to symmetric pulses. Rather, pulses of any arbitrary intensity shapes and phase shapes can be analyzed.
As stated in the foregoing, the method according to the present invention method measures ultrashort optical pulses and can measure the intensity shape and phase shape of pulses by using a compact and less expensive measuring system with a precision higher than conventional methods. As the techniques of this invention can obtain the pulse phase in extreme detail, it can be effectively applied in the measurement of optical pulses before and after passing through an optical fiber transmission line to observe the intensity and phase shapes thereof. This allows a direct and simplified evaluation of characteristics of optical fiber lines. Moreover, this invention is generally applicable in optical systems and optical materials.
This invention which allows detailed observation of pulse phase provides an effective means in optimization and adjustment of pulse compressors with optical fibers.
This invention is a basic method applicable in a wide variety of fields concerning generation, formation, propagation, etc. of ultrashort pulses.